Elliptic Curves x³ + y³ = k of High Rank

2004/03/06 by Noam D. Elkies, Elkies, Noam D., Nicholas F. Rogers +1
Mathematics · #11G05 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G05

paper · pdf · doi:10.48550/arxiv.math/0403116

10 pages; to appear in the Proceedings of ANTS-VI (Algorithmic Number Theory Symposium, 2004)

arxiv created 2004/03/06 · arxiv updated 2009/12/01

Abstract

We use rational parametrizations of certain cubic surfaces and an explicit formula for descent via 3-isogeny to construct the first examples of elliptic curves E_k: x³ + y³ = k of ranks 8, 9, 10, and 11 over Q. As a corollary we produce examples of elliptic curves xy(x+y)=k over Q with a rational 3-torsion point and rank as high as 11. We also discuss the problem of finding the minimal curve E_k of a given rank, in the sense of both |k| and the conductor of E_k, and we give some new results in this direction. We include descriptions of the relevant algorithms and heuristics, as well as numerical data.

Related

• Two infinite families of elliptic curves with rank greater than one
• On the rank of certain parametrized elliptic curves
• Three lectures on elliptic surfaces and curves of high rank
• Infinite family of elliptic curves of rank at least 4
• On the ranks of elliptic curves with isogenies